1. Radius Patterns follow Grid Lines
The ring version of Wall of Fire says this about its area of effect:
a ring of fire with a radius of up to 5 ft. per two levels
If you turn to the back of the DMG (p. 307) or look at this answer, you can see some patterns for radius spread effects. All of them spread along square edges. They're a ring of squares, rather than a circle. In this pattern, a square can't be half in or half out, because it only travels along the lines.
(This is actually also true for line effects, as PHB p. 176 shows. If a line goes through any part of a square, the entire square is affected. For something like Wall of Fire, an entire square has to be in one state or the other to follow with this, but if you place it along the grid lines it's fairly easy to do that.)
2. Yes
As you noted, Blade Barrier doesn't give a thickness or anything to imply that it's more than a curtain of blades (in fact, it says its a curtain of blades). You can treat it the same way as Wall of Fire: draw it on the grid lines. (The rules for area of spells mention that you should draw it starting from a grid intersection, so you can follow the grid lines.)
Regardless of the shape of the area, you select the point where the
spell originates, but otherwise you don’t control which creatures or
objects the spell affects. The point of origin of a spell is always a
grid intersection. When determining whether a given creature is within
the area of a spell, count out the distance from the point of origin
in squares just as you do when moving a character or when determining
the range for a ranged attack. The only difference is that instead of
counting from the center of one square to the center of the next, you
count from intersection to intersection.
This also makes the cover part of it easy to handle: anything on one side of that grid line attacking the other side has to deal with the cover portion of the spell.
3. Creature Size Matters
If you're placing it on grid lines (as you should), then yes, you can't actually place it on top of a medium creature. Large (or bigger) creatures take up multiple squares and you could have the barrier cast such that it appears in the middle of one. That makes this part of the spell make sense:
If you evoke the barrier so that it appears where creatures are, each
creature takes damage as if passing through the wall. Each such
creature can avoid the wall (ending up on the side of its choice) and
thus take no damage by making a successful Reflex save.
Medium or smaller creatures wouldn't be damaged, as the barrier is not "where the creatures are".
4. Because the spell says so
This point has no rules-as-written answer, except because the spell says so. :)
I'd speculate that the game designers did that with the reasoning that as the spell comes into existence, someone can react and get out of the way before it can fully attack. Whereas once it's already there, someone has to go through it. But I don't know of any rules, anywhere, that explain why they did it this way.
5. Yes, you can attack
Blade Barrier says this:
Any creature passing through the wall takes 1d6 points of damage per
caster level (maximum 15d6), with a Reflex save for half damage.
By a strict reading, passing through the wall is movement. By that reading: yes, you can attack through it without taking damage. I'm not aware of any rules clarification on that point.
Anybody attacking through it does have to deal with cover:
A blade barrier provides cover (+4 bonus to AC, +2 bonus on Reflex
saves) against attacks made through it.
A wall of force grants cover by being an obstacle. A confirmed tweet from a game designer states this includes spells.
According to the cover rules in the Player's Handbook, p.196:
Walls, trees, creatures, and other obstacles can provide cover during combat, making a target more difficult to harm. [...] A target has half cover if an obstacle blocks at least half of its body [...] A target has has three-quarters cover if about three quarters of it is covered by an obstacle [...] A target has total cover if it is completely concealed by an obstacle.
Is a wall of force an obstacle? In D&D 5th edition, words not defined in game rules are interpreted according to their standard English meaning.
Something that impedes, stands in the way of, or holds up progress.
The description of the wall of force states:
Nothing can physically pass through the wall.
Therefore, the wall of force is an obstacle, and being behind it relative to your opponent grants you cover.
If I understand your recent questions correctly, you hope to learn whether wall of force specifically blocks magic, on the hypothesis that magic is not "physical". The rule on casting spells against people behind obstacles appears in the Player's Handbook, page 204:
To target something, you must have a clear path to it, so it can't be behind total cover.
D&D 5e designer Jeremy Crawford, whose rulings are considered official, confirms in a tweet that wall of force provides total cover:
Q: could a wizard make a sphere around a creature using wall of force and then chill touch to damage them through the wall?
Crawford: Unless a spell says otherwise, you can't target someone behind total cover (PH, 204)
Also here, in specific reference to wall of force:
Cover is a physical obstruction, not necessarily a visual one.
While the cover rules say that total cover "completely concealed", a term which in earlier editions of D&D referred specifically to visibility and not cover, Crawford here appears to clarify that "concealed" is a synonym here for "covered". Note how the other two forms of cover say "blocks" and "covered", so it seems that they're using synonyms to avoid repetition.
Therefore, regardless of the "physicality" of magic, the official ruling on this issue is that wall of force provides total cover, and therefore blocks magic.
Best Answer
You will never have to worry about passwall targeting the barrier
Passwall's description states (emphasis mine):
The surface of the Cube of Force creates "a barrier of Invisible force". If the caster cannot see the barrier, typically requiring a spell like See Invisibility, then they cannot target the barrier. But even if they can see it, the barrier isn't wood, plaster, or stone. Thus, it is an invalid target for Passwall as far as a direct cast.
With that in mind, the only thing we need to worry about is the cube's barrier overlapping with an opening, created by Passwall, on a different surface. In that case, Passwall's effect has nothing to do with opening a passage in anything other than the surface it was originally cast on. Thus, we don't have to worry about any effect applying to the cube, aside from the specific description in the cube's text.
The cube loses 1d6 charges because the item's description says it does, but no hole is created in the barrier because the cube didn't say that would happen, and the Passwall spell wasn't cast on the barrier.
Repeatedly colliding with a Wall of Fire will continue to drain charges
The cube's description only states:
The text makes no mention of "The first time the barrier is targeted..." in contrast to many other effects that do use this language. For example Wall of Fire itself, which says:
From this, we can determine that no matter how many times you run into the same Wall of Fire, the cube will continually lose charges, because there is no exception which states that the drain was only meant to be applied one time.